Abstract

Bistable responses of Fabry–Perot cavities and optical arrays in the presence of diffraction and diffusion are studied both analytically and numerically. The model is a pair of nonlinear Schrödinger equations coupled through a diffusion equation. The numerical computations are based on a split-step method, with three distinct characteristics. In these diffusion-dominated arrays with weak diffraction, this study demonstrates that focusing nonlinearity can improve the response characteristics significantly. The primary results of the study are that (1) for diffusion-dominated media a small amount of diffraction is sufficient to alter optical bistability significantly; (2) focusing nonlinearities enhance optical bistability in comparison with defocusing nonlinearities; (3) in diffusion-dominated media these focusing–defocusing effects are quite distinct from self-focusing behavior in Kerr media; (4) in the presence of diffraction the response of the array can be described analytically by a reduced map, whose derivation can be viewed as an extension of Firth’s diffusive model to include weak diffraction; (5) this map is used to explain analytically certain qualitative features of bistability, as observed in the numerical experiments; and (6) optimal spacing predictions are made with a reduced map and verified with numerical simulations of small all-optical arrays.

Hysteresis curves of a diffusive cavity for inputs of various widths. Left-most curve, a δ-function input. (Throughout the paper, computed points are linked by straight lines to construct curves for ease of visualization.)

Adiabatic solutions that mimic hysteresis loops. In each case the power Pin of the Gaussian input is linearly (in time) ramped up from 0 to 2.5 during a time interval of 500 round trips and then linearly ramped down to 0 during the same period of time. The solid curve has self-focusing nonlinearity, and the dashed curve has self-defocusing nonlinearity, both with Γ=0.005. Dotted curve, diffractionless.

Solutions of a self-focusing cavity for different levels of diffusion. The input is distributed as a hyperbolic secant function: Fin=(Pin/2w)1/2sech(x/w), where the width w=0.9 and the power Pin=2.20.Γ=0.02 for all curves.